An Empirical Relationship Between Interplanetary Conditions and Dst

R. K. BURTON, R. L. MCPHERRON, AND C. T. RUSSELL

Institute of Geophysics and Planetary Physics, University of California Los Angeles, California 90024

Originally Published In:J. Geophys. Res., 80 (31), 4204-4214, 1975.

Abstract. An algorithm is presented for predicting the ground-based Dst index solely from a knowledge of the velocity and density of the solar wind and the north-south solar magnetospheric component of the interplanetary magnetic field. The three key elements of this model are an adjustment for solar wind dynamic pressure, an injection rate linearly proportional to the dawn-to-dusk component of the interplanetary electric field which is zero for electric fields below 0.5 mV m-1, and an exponential decay rate of the ring current with an e folding time of 7.7 hours. The algorithm is used to predict the Dst signature of seven geomagnetic storm intervals in 1967 and 1968. In addition to being quite successful, considering the simplicity of the model, the algorithm pinpoints the causes of various types of storm behavior. A main phase is initiated whenever the dawn-to-dusk solar magnetospheric component of the interplanetary electric field becomes large and positive. It is preceded by an initial phase of increased Dst if the solar wind dynamic pressure increases suddenly prior to the main phase. The recovery phase is initiated when the injection rate governed by the interplanetary electric field drops below the ring current decay rate associated with the ring current strength built up during the main phase. Variable recovery rates are generally due to additional injection during the recovery phase. This one algorithm accounts for magnetospheric behavior at quiet and at disturbed times and seems capable of predicting the behavior of Dst during even the largest of storms.

INTRODUCTION

The magnetosphere responds to the solar wind driving function in a variety of complex ways, this behavior producing various types of disturbances. These disturbances can be monitored on the earth to provide estimates of the level of magnetospheric activity. Dst, computed from mid-latitude ground magnetograms, is one widely used parameter.

Geomagnetic storms, as seen in Dst, commonly have three phases: a sudden commencement, a main phase, and a recovery. However, other than the fact that sudden commencements often precede the main phase, there is little relationship between them. Piddington [1963] noted that the size of sudden commencements was independent of the main phase minimum. Hirshberg [1963] found evidence for ring current enhancement without sudden commencements. Akasofu [1964], in addition, found that sudden commencements are not always followed by storm main phases or auroral activity.

Qualitatively, the relation between the various phases of a storm and the solar wind is becoming understood. It has generally been accepted that sudden commencements are associated with enhancements of solar wind dynamic pressure. Burlaga and Ogilvie [1969] found that sudden commencements were associated with hydromagnetic shocks in the solar wind. Empirically, the size of the sudden commencement was found to be proportional to the square root of the solar wind dynamic pressure [Siscoe et al., 1968; Ogilvie et al., 1968].

The Z component of the interplanetary magnetic field (IMF) has been associated with geomagnetic activity in general [Hirshberg and Colburn, 1969; Arnoldy, 1971; Foster et al., 1971] and the geomagnetic storm main phase in particular [Rostoker and Fälthammar, 1967; Russell et al., 1974]. Rostoker and Fälthammar [1967] found that the storm main phase was associated with a sustained southward Bz. Russell et al. [1974] found that the southward Bzhad to exceed an apparent threshold level, possibly Dst-dependent, in order to trigger a storm main phase. Rostoker and Fälthammar [1967] also noted that the recovery phase was associated with a decrease or switching off of the southward Bz. Davis and Parthasarathy [19671 observed that the rate of recovery was related to the magnitude of Dst.

Thus while quantitative information exists on the relationship both between sudden commencements and dynamic pressure and between the recovery phase and Dst, the main phase decrease is only qualitatively understood in terms of' the southward component of the IMF. This dependence is believed to be the result of an enhanced merging rate between the IMF and the geomagnetic field when the IMF is southward. The dependence of the merging rate on the southward component and the rate of ring current injection upon the merging rate have not been found.

This paper presents an empirical relationship for the rate of change of Dst in terms of the dawn-dusk solar wind electric field (the solar wind velocity times the north-south IMF) and the solar wind dynamic pressure. The rate of ring current injection is expressed in terms of the dawn dusk solar wind electric field.

ANALYSIS

The solar wind parameters used in this study to compute the solar wind dawn-dusk electric field and dynamic pressure were obtained from solar wind velocity and density data measured on Explorer 33 and 35 provided by J. Binsack and corresponding IMF data from D. S. Colburn and N. F. Ness. The temporal resolution of the velocity and density data supplied was 163 s, while that of the magnetic field data was 82 s. All of the interplanetary data were interpolated to provide compatible data sets of 2.5-min resolution. Interpolation was performed only over contiguous data and not across data gaps exceeding 350 s. From these data the solar wind dawn-dusk electric field (E = VBz 10-3mV/m) and dynamic pressure (P = nV2 10-2 eV/cm-3) at the satellite position were computed (V in kilometers per second, n in units per cubic centimeter, and Bzin gammas).

Dst was determined from the digitized H components of the 11 mid-latitude stations given in Table 1, supplied by the World Data Center. Dst was determined with 2.5-min resolution by Fourier-fitting the difference between the storm and quiet day H values, using F = C + Σn=13(ansin 2πnt/T + bncos 2πnt/T where T = 24 hours. Dst is then equal to the constant C.

Data from 1967 and 1968 were used in this study. There were about 85 storms during these 2 years with Dst < -40 γ. None of these storms were arbitrarily excluded. However, the requirement of simultaneous solar wind velocity, density, and magnetic field data from the same satellite with no large data gaps is quite severe. Often neither Explorer 33 nor Explorer 35 was in the interplanetary medium. As a result, the seven storm intervals listed in Table 2, consisting of 10 geomagnetic storms, provide the basis of this study. In addition to the storm days given in Table 2, the corresponding quiet days used in the computation of Dst are given for each storm sequence.

THE ALGORITHM

Dst is a measure of the worldwide deviation of the H component at mid-latitude ground stations from their quiet day values. The H component at mid-latitudes on a quiet day as well as on disturbed days is a function of magnetopause currents, the ring current, and tail currents. Dst is determined by averaging the records from N mid-latitude stations in the following manner:

a local time average of the difference between Hdisturbed and Hquiet. We can rewrite this as

where the superscripts d and q refer to disturbed and quiet, respectively. The contributions of the various currents can be explicitly written as

where the subscripts mp and rc refer to magnetopause and ring current, respectively (the tail currents are included with the ring current here). Since averaging is a linear operation, we may rewrite this as

We can solve for the effect of the disturbance ring current:

Defining and noting that the contribution of magnetopause currents to H is proportional to the square root of the solar wind dynamic pressure, experimentally confirmed by Siscoe et al. [1968], Ogilvie et al. [1968], Verzariu et al. [1972], and Su and Konradi [1975], and that the quiet day terms are constants determined by the quiet day magnetopause and ring currents, we have

Dst0 = Dst - b(Pd)1/2 + c (2)

During the recovery phase of a geomagnetic storm, when the ring current is symmetric, the Sckopke relation [Sckopke, 1966] can be used to find the total energy in the ring current from Dst0. During the main phase, when the ring current is asymmetric, a more complex relation involving the radial distance and longitudinal extent of the current would be necessary. However, even when the ring current is asymmetric, Dst0 gives the total energy in the ring current to better than 20% [Siscoe and Crooker, 1974].

Thus, given the solar wind dynamic pressure and having determined the constants b and c, one could predict Dst if one could predict the strength of the ring current Dst0. It is reasonable to assume following the work of Davis and Parthasarathy [1967] that the rate of decay of the ring current is proportional to the strength of the ring current in the absence of sources. In this case then, (d/dt)Dst0 = -aDst0, where Dst0 < 0 by definition.

We will show that such a simple form is consistent with the data even though there are a variety of ring current dissipation mechanisms which possibly all occur with varying strengths for different storms and different phases of the storm.

Finally, changes in Dst0 will also occur as a result of ring current injection. Although we know that the solar wind interaction with the magnetosphere is exceedingly complex, we will approximate the rate of energy input to the ring current by a function of only the interplanetary electric field. The historical justification of this choice is found in the work of Rostoker and Fälthammar [1967], Arnoldy [1971], Russell et al. [1974], etc. While other parameters may have some effect, it must be small with respect to that of the electric field. The ultimate test of this choice, of course, lies in the success of the correlation itself. Thus, in the presence of injection of energy the rate of change of Dst0 is

where F(E) < 0 for injection. This equation has both a threshold for a net ring current increase dependent on ring current strength and a saturation ring current for a given rate of energy input. The problem at hand is to find the optimum choice for F(E) and the constants a, b, and c.

By using a prime to denote measurements at the nose of the magnetosphere, (3) can be rewritten in terms of the experimental quantities by use of (2):

Physically, changes in Dst have three causes. The first term on the right describes the ring current decay proportional to the ring current, the second term describes ring current injections dependent upon the solar wind electric field, and the third term describes changes in Dst resulting from changes in the magnetopause currents. We note that E'(t) and P'(t) can only be computed from solar wind density, velocity, and magnetic field data measured at the satellite and not at the nose of the magnetosphere as they should be. To obtain the desired quantities, we must further introduce a time delay (Tsw) to account for the travel time from the satellite to the magnetopause and any further delays introduced by the magnetosphere itself (Tm). Further, we note that the magnetospheric response may be frequency-dependent or, in other words, the interplanetary electric field may be attenuated by a transfer function H(ω).

DETERMINING THE OPTIMUM PARAMETERS

In order to generate Dst from (4), a number of quantities must be empirically determined. These are as follows:

a

measure of ring current decay;

b

measure of the response to dynamic pressure changes in the solar wind;

c

measure of the quiet day currents;

tm

response time, insofar as ring current injection is concerned, of the magnetosphere to an applied electric field;

H(ω)

frequency response of the magnetosphere to an applied electric field;

F(E)

rate of ring current injection as a function of a delayed and filtered solar wind electric field.

In general, ring current injection and decay, as well as changes in the solar wind dynamic pressure, can occur at the same time. This makes the determination of the above parameters difficult. However, intervals can be chosen in which only one process is dominant, and thus the determination of the associated parameter is made much easier. To this end, it is of interest to discuss the general properties and asymptotic behavior of (4). First, if F(E) = 0 (which apparently occurs for the dusk-dawn electric field) and the disturbed dynamic pressure were steady but not equal to the quiet dynamic pressure, then

This is often the case during the recovery of a geomagnetic storm, where Dst < 0. In this case, Dst would increase, and the ring current would decay until (d/dt)Dst = 0, when

Dst = b(Pd)1/2 - b(Pq )1/2 + |Hrcq| (5)

In other words, Dst would decay, not to zero but to a value determined by the difference between the disturbed and quiet day dynamic pressures and the ring current on the quiet day. if Pd = Pq, Dst would decay to a positive value determined by the effect of the quiet day ring current. If the ring current on the quiet day were zero, then with Pd = Pq, Dst would decay to zero.

If, instead, there were a constant dawn-dusk electric field (F(E) = I),an idealization of the situation during the main phase of a geomagnetic storm, and Pd = Pq= const for simplicity, then (4) would be

In other words, Dst would asymptotically approach a level determined by the rates of injection and decay of the ring current.

If F(E) = 0 and there were a step change in the dynamic pressure from Pq to Pd, Dst would jump from Dst =|Hrcq| (from (5)) to Dst = b(Pd)1/2 - b(Pq)1/2 + |Hrcq| . Dst would remain at this new level as long as Pdwere constant and F(E) = 0. This idealizes the situation for a sudden commencement.

In the following sections we discuss the determination of the above parameters individually.

Changes in dynamic pressure. During a sudden commencement, Dst should increase by the amount bΔ(P)1/2, where P is the solar wind dynamic pressure. Since P is measured, it can be simply determined. This has been done previously by Siscoe et al. [1968], Ogilvie et al. [1968], Verzariu et al. [1972], Su and Konradi [1975], obtaining b = 0. 14, 0.17, 0.35, and 0.43 γ(eV cm-3)-1/2, respectively.

Three sections of data in our sample (sudden commencements on February 15, 1967; February 23, 1967; and February 15, 1968) permitted a determination of this constant in the absence of complicating factors. The average slope determined from ΔDst/Δ(P)1/2is b = 0.2 with a standard deviation of 0.1, in reasonable agreement with previous measurements.

Quiet day constant. Late in the recovery phase of a storm, where P = const and F(E) = 0, Dst will asymptotically approach a value given by Dst -b(P)1/2 + c = 0. Su and Konradi [1975]chose quiet days (they did not have electric field information) and plotted Dst versus (P)1/2 . Both constant b and constant c can be determined in this way. Su and Konradi found that b = 0.2-0.4, in agreement with their other determination of b, and c = 19 γ. Our data are consistent with this value, and we have used c = 20 γ in our work. Since c = b(Pq)1/2 + Hrcqand b(Pq)1/215 γ, the field of the quiet day ring current is probably less than 5 γ.

Ring current decay. Parameters b and c being determined, Dst0can be computed from Dst by using (2). During the recovery phase of a storm, for intervals when there is no ring current injection, the ring current decay constant can be evaluated from (d/dt)Dst0 = -aDst0.Previous studies [cf. Kokubun, 1972; Russell et al., 1974] have indicated that ring current injection occurs for southward IMF's. Therefore the ring current decay constant was computed from thirty-two 1-hour intervals during storm recoveries when there was northward IMF. The results are given in Figure 1, where the rate of decay is plotted versus Dst0. Our parameter 'a' is the negative of the slope of this plot. There is much scatter in this display in part because of errors in our determination of Dst0and in part because of uncertainties in measuring the decay rate. Thus we have used only points with Dst0 < -20 γ to determine the average slope, shown by the center line in this figure. The slope of this line is -0.13 hour-1, which corresponds to a decay time of' 7.7 hours.

The upper and lower lines are lines with slopes one standard deviation greater and less than the average. Most of the points not bounded by these lines have small values of decay rate and Dst0. Again, these are the points with the greatest percentage errors. In drawing a straight line, i.e., using the average slope, we have assumed that the slope is independent of Dst0. The data suggest that this may not be true, for the curve appears to flatten at high Dst0. However, we feel that we have insufficient data to pursue this point.

Although other workers have studied the ring current decay rate, other work on ring current decay suffers because it is not known when injection occurs, especially injections that simply slow the recovery. To overcome this, Davis and Parthasarathy [1967] assumed little injection when DP activity was low. Using 3-hour decreases in Dst, they obtained expressions for the decay which differ in form from ours but which give very similar decay rates as a function of Dst. Their decay rates are somewhat slower than ours, presumably because some injection existed during their intervals. Another way to combat the lack of solar wind data is to find the maximum decay rate. In such a study, Shevnin [1973] found a maximum decay rate essentially identical to our rate.

Fig. 1. The decay rate observed for 30 intervals of northward interplanetary field as a function of Dst normalized to a constant dynamic pressure.

Magnetospheric response to E. In general, we anticipate that the magnetospheric response to an impressed solar wind electric field, as reflected in Dst, can be divided into three parts: the response time Tm to the applied electric field, e.g., time delays inherent in magnetospheric processes; the frequency response H(ω), e.g., the averaging out of rapid oscillatory changes; and the functional dependence of the ring current injection on the electric field F(E). The response time of AE with respect to a southward Bz has been determined by correlation analysis by Rostoker et al. [1972] and Meng et al. [1973] as 30-50 min and 10-90 min, respectively. These times are consistent with Arnoldy's[1971] 1-hour delay, using 1-hour resolution data. These results suggest the possibility of a similar delay associated with Dst. In addition, since physically a delay or response time could simply reflect the presence of an attenuated high-frequency response, we might anticipate that the transfer function H(ω) might become small at high frequencies corresponding to periods much shorter than this delay time.

Given the solar wind dynamic pressure, the constant b determining the effect of magnetopause currents, the constant c representing quiet day currents, and the ring current decay constant a, ring current injection can be determined in terms of the electric field, i.e., F(E). We found no clear injections except when Eyin GSM coordinates was positive. This corresponds to a southward IMF, or a dawn-to-dusk electric field.

Thus, to determine the response time to a change in Ey, we used intervals in which Eyswitched from negative or slightly positive to a large positive value or vice versa. In these cases the rate of ring current injection changes abruptly, and the delay between the change in electric field at the nose of the magnetosphere and the response in Dst can be determined. This procedure is only valid in the absence of dynamic pressure fluctuations, which often are associated with rapid electric field changes. As a result, only four cases are ideally suited to determine tm. These give an average value of 25 min, there being a variation from 0 to 40 min. This is similar to the AE response time, though somewhat smaller.

The last remaining task is to compute F(E). To do this, we selected intervals of at least 1/2-hour duration in which the dynamic pressure was constant. Then we measured the rate of change of the ring current and compared it with the Y solar magnetospheric component of the interplanetary electric field, i.e., the product of the solar wind velocity and the north-south component of the magnetic field. The results are shown in Figure 2. These data clearly show the rectification of the interplanetary electric field by the magnetosphere. The straight line on the right-hand side is the linear least squares fit to the values for positive Ey, i.e.,

F(E) = 1.26 X 10-3 Ey + 1.75 x 10-4

where F(E) is expressed in gammas per second and E in millivolts per meter.

While the least squares linear fit very nearly passes through the origin, it is not clear from the data that the origin is the center about which the rectification takes place. In fact, it appears from these data that for low injection rates the true curve is steeper than the least squares fit. Figure 3 shows these data on a log-log plot. While it is apparent that the best approximation to the overall curve is linear in the electric field, at small electric field strengths the dependence on the electric field appears to strengthen. Since most injection events occur for low to moderate electric field strengths, we have approximated this behavior by slightly steepening the slope of our linear approximation to F(E) and forcing it to intercept the F(E) = 0 level for positive Ey (i.e., at Ey = 0.5 mV m-1). Our confidence in the appropriateness of this modification to the least squares fit is strengthened by its success in modeling the more extended intervals of data shown in later sections, which for reasons of their temporal variability were not included in Figures 2 and 3.

Fig. 2. The ring current injection rate observed for 23 intervals of approximately constant dynamic pressure as a function of the dawn-to-dusk component of the IMF. Injection rates have been corrected for the intrinsic rate of decay of the ring current measured during periods of no injection. The line in the right-hand panel is the linear least squares fit to the points in the right-hand panel.

Finally, we note that rapidly oscillating electric fields were found to be not as efficiently rectified as slowly varying fields. To incorporate this behavior in our algorithm, we filtered the data with a low-pass filter with a corner frequency of 2 cph and an attenuation of 6 dB per octave, after testing filters with corner frequencies of 5 min to 2 hours on periods with particularly strong high-frequency oscillations in the electric field. This filter corresponds to the H(ω) discussed above.

The final prescription. Equation (4), with the empirical parameters discussed above, is

The asterisk denotes convolution, and h(t) is the impulse response of the magnetosphere, corresponding to H(ω) in the frequency domain.

Application of the prescription. Using the prescription, we have generated Dst for the storm intervals given in Table 2. In Figures 4-10 we have plotted the solar wind electric field, the square root of the solar wind dynamic pressure, Dst determined from ground magnetograms, and Dst determined by (4). All of the quantities are plotted to 2.5-min resolution. Equation (4) was initialized by use of the initial Dst from ground magnetograms for each interval. The storm beginning February 23, 1967 will be discussed first, because it is the simplest one containing all of the features in a sudden commencement storm. The remaining storms will be discussed in chronological order.

February 23 and 24, 1967. Initially, as shown in Figure 4, at 0000 UT on February 23, Dst = - 10 γ. During the first few hours, E, oscillated between small negative and positive values, small amounts of ring current injection being produced when E, was positive. At the same time the solar wind dynamic pressure also fluctuated in a periodic fashion. This dynamic pressure variation instantaneously produced the oscillations in Dst at this time.

Subsequently, the electric field was negative, no injection thus being provided, and the dynamic pressure was nearly constant. Thus the decay in Dst is simply the result of ring current decay.

Between 0800 and 1200 UT the dynamic pressure increased to a maximum. During this interval there were three positive spikes in E, which because of their short duration produced little ring current injection and hence little effect on Dst. The effect of the dynamic pressure being dominant, Dst increased at this time to a maximum. Similarly, the decrease in Dst immediately following 1200 UT resulted from the decrease in dynamic pressure.

Between 1300 and 1900 UT a continuous positive E, produced ring current injection which drove Dst to a main phase minimum near -60 γ.

After 1900 UT, Eywas mostly negative, except for small positive values around 0800 UT on February 24, and the dynamic pressure was virtually constant. As a result the storm entered a sustained recovery phase. Note that the onset of the recovery phase occurred while there still existed a positive Ey, though its small magnitude did not provide enough injection to overcome the rate of ring current decay. The Dst index determined from ground observations deviates from the prediction beginning 0400 UT on February 24 in that it exhibits a slower decay rate. We will discuss this later.

Briefly, a pulse of enhanced dynamic pressure produced the sudden commencement peaking at 1200 UT on February 23. The following interval of positive Ey generated the main phase of the magnetic storm. A reduction in positive Ey, and hence in ring current injection, initiated the recovery phase. The subsequent continued recovery was allowed by the nearly continuous negative Ey, which provided virtually no further ring current injection.

Fig. 3. A log-log plot of the positive injection rates as a function of electric field for the right-hand panel of Figure 2. Lines of slopes 1 and 2 have been drawn to illustrate the better overall agreement with a linear dependence on electric field.

Fig. 4. (From top to bottom) The square root of the solar wind dynamic pressure, the dawn-to-dusk component of the interplanetary electric field. and the predicted (dashed line) and observed (solid line) Dst for February 23 and 24, 1967. The electric field has been plotted with positive values below the axis to emphasize the correspondence between southward interplanetary fields and dawn-to-dusk electric fields and because Dst injections produced by dawn-to-dusk fields cause the Dst trace to decrease.

February 7 and 8, 1967. At 0000 UT on February 7, as shown in Figure 5, Dst was close to zero. A nearly constant negative Ey was present until 1500 UT, while the dynamic pressure slowly decreased. Dst, with no ring current injection and nearly constant dynamic pressure, remained near zero.

Beginning about 1500 UT, the dynamic pressure increased, this behavior tending to raise List. During the time of increasing dynamic pressure, Eybecame positive and thus ring current injection was produced, this event tending to decrease Dst. Injection dominated during this interval, except for the sharp increase in dynamic pressure just prior to 1700 UT, and thus Dst decreased.

The subsequent main phase decrease was generated by the positive Ey from 1800 to 2300 UT, slightly enhanced by the decreasing dynamic pressure over this interval. Relatively little injection occurred between 2300 and 0300 UT oil February 8. As a result, decay dominated, and Dst recovered. Slightly enhanced injection produced a quasi-balance between injection and decay around 0500 UT.

After 0700 UT, Ey was positive for 7 hours, providing the ring current injection to produce another Dst minimum. Following 1400 UT, Eywas negative, and thus List could recover.

Between 1800 and 2100 UT the increased rate of recovery in Dst and subsequent decrease were the result of' the similar dynamic pressure variation during this interval.

February 15, 16, and 17, 1967. As is shown in Figure 0. since Ey was near zero and the dynamic p ressure constant. Dst remained near zero until a few minutes before 0000 UT oil February 16. Between 0000 and 0800 UT, with no ring current injection as a result of a negative Ey, the variation in Dst was produced by tile changes in dynamic pressure. Between 0800 and 1000 UT, injection produced by positive Eycompeted with dynamic pressure produced variations. Subsequently, ring current injection dominated and drove the main phase decrease front 1000 to 1300 UT.

During the initial part of the storm recovery phase the rate of recovery was somewhat enhanced by decreasing dynamic pressure near 1400 UT and retarded by additional ring Current injection between 1400 and 1900 UT

After 1900 UT, Ey was near zero. and the dynamic pressure was low and nearly constant. As was true for the decay on February 24, 1967. the ground-determined Dst deviated from our predicted Dst in that the decay of ground Dst again was slower.

February 15 and 16, 1968. At 0000 UT on February 15, as is shown in Figure 7, Dst is slightly greater than zero. The Dst variation over the first 7 hours was due primarily to changes in the dynamic pressure, since the electric field was negative for virtually the whole time. The electric field was extremely variable on February 15.

Following the sudden commencement at about 0800. produced by the dynamic pressure enhancement at that time. ring current injection was driven by intervals of' dawn-to-dusk field from 0800 to 1800 UT, The predicted decrease does not match tile actual decrease in ground Dst, partly because the effect of dynamic pressure was somewhat overestimated, as is indicated by the sudden commencement. Another possibility is the high variability in the electric field. Qualitatively, the Dst variation is predicted correctly.

After the drop at 1500 UT the dynamic pressure remained nearly constant. Since there was little or no injection from 1800 to 0900 UT on February 16, Dst recovered. Injection produced by westward fields from 0900 to 1500 UT resulted in a slight decrease in Dst during that time. Recovery again occurred during a period of negative field. The short amount of positive field at the end of February 16 produced minimal injection which had little effect on Dst.

February 27, 28, and 29, 1968. At 0000 UT on February 27, as shown in Figure 8, Dst = 0γ. Ey is approximately zero until 1300 UT. The dynamic pressure being approximately constant, Dst changes very little. The small positive field from 1300 to 1400 UT produced little injection and hence little effect on Dst. The dynamic pressure began to increase at about 1300 UT, this behavior causing the increase in Dst through 1800 UT.

The electric field went positive from 1800 through 0200 UT on February 28. Dst began to decrease shortly after 1800 UT and reached a minimum of -30 γ around 0100 UT. Between 0200 and 1200 UT there was no positive field. This and the enhanced dynamic pressure drove Dst upward from 0100 to 1200 UT, Dst becoming positive between 0900 and 1200 UT.

After 1200 UT the electric field was predominantly positive, its magnitude diminishing erratically through the end of February 29. At the same time the dynamic pressure was decreasing. Dst decreased to a minimum of -50 γ, while (4) produced a somewhat lower minimum of -70 γ. This contrasts with the February 15, 1968, storm, where the ground Dst minimum exceeded that produced by (4).

The ground Dst stayed fairly constant around -40 to - 50 γ over the first 12 hours of February 29. Then, since the diminishing positive electric field was no longer able to overcome the ring current decay, Dst began its recovery. The small decrease occurring after 1800 UT was due to the positive field from 1800 to 1900 UT. On the other hand, this occurrence allowed the empirically generated Dst to stabilize at the level to which the actual Dst was driven. Thus, while injection appears to be overestimated by the empirical expression from 2200 UT on February 28 to 1800 UT on February 29, the injection around 2000 UT on February 29 matched the ground Dst quite well.

March 3, 4, and 5, 1968. Throughout this 3-day sequence the electric field was nearly always positive. During this interval, as is shown in Figure 9, Dst was maintained at an approximately steady balance between ring current injection and decay. At the beginning of March 3, Dst = -30 γ, and it is recovering slowly with Ew 0. Small positive electric fields from 0700 through 1500 UT maintained Dst between -20 and -30 γ. A relatively larger positive field between 1500 and 1800 UT drove Dst down to about -50 γ near 1900 UT. A diminished but still positive field allowed only slow recovery until another relative enhancement in positive field caused a drop in Dst near 1000 UT on March 4. This sequence is repeated with a recovery from 1000 to 1400 UT due to reduced positive field and Dst decreases from 1400 to 1700 UT due to enhanced positive field.

Until about 2200 UT the dynamic pressure was constant. Between 2200 and 0200 UT on March 5, the pressure increased. This, coupled with a reduced positive electric field, resulted in an increase in Dst from 2200 to 0700 UT on March 5. About 0730 UT the dynamic pressure dropped abruptly. At the same time the electric field went to a relatively large positive value, and it remained there until about 1200 UT. These two changes produced the Dst decrease from 0700 to 1200 UT to -60 γ. The reduced positive field after 1200 UT allowed Dst to recover somewhat after that.

May 1 and 2, 1968. As is shown in Figure 10, Dst is initially slightly negative on May 1. A negative electric field causes Dst to recover gradually. A slowly increasing positive field beginning at about 1200 UT produced a main phase minimum of -50 γ at about 2200 UT. The dynamic pressure was constant the whole time.

Beginning at 2200 UT the positive electric field diminishes to zero, while the dynamic pressure increases to a peak slightly before the day change. The sudden commencement that this behavior would produce is lost in the start of the recovery phase of the storm. The lack of significant injection on May 2 coupled with the somewhat higher dynamic pressure produced the recovery of Dst.

DISCUSSION

Dst is a quantity derived from mid-latitude ground magnetograms. It is a function of currents inside and on the boundary of the magnetosphere. Like other measures of geomagnetic activity it is a measure of energy transfer from the solar wind to the magnetosphere. Its magnitude is inevitably a function of solar wind properties, the state of the magnetosphere, and the physical processes involved in the solar wind-magnetosphere interaction. In (4) we have attempted to relate the currents which produce Dst to solar wind parameters. The constants and functional relationships that were empirically determined represent various multiple and complex interactions between the solar wind and the magnetosphere as well as within the magnetosphere. These processes are not well understood at this time. Almost certainly, the constants and functional relations have sonic dependence upon the state of the magnetosphere and therefore on its previous history. An assumption in the use of (4) is that this dependence is relatively small and that, insofar as Dst is concerned, the magnetosphere can be treated as being continuously in an average state, with its physical interactions unchanging. A priori, this is not necessarily valid. However, since Dst derived from (4) agrees well with Dst derived from ground magnetograms, the assumption appears to be reasonably valid.

It should be noted that the deviation of the empirical Dst from the ground Dst is not a simple measure of the validity of (4) or the assumptions that went into it. Dst determined from ground magnetograms is itself an approximation to the quantity (4) is attempting to produce. As a result, deviations of Dst determined by (4) from ground Dst may be due, in one extreme, simply to the deviation of ground Dst from the ideal Dst.

Computation of an ideal Dst would require more complete equatorial ground station coverage and a better definition of the quiet day. In this particular study, the ground station coverage, as indicated in Table 1, is reasonably good and should not result in much error. However, the choice of a quiet day in the computation of Dst is a problem. Very simply, truly quiet days are rare occurrences. As a result, some approximation to a quiet day must be used. Two reasonable choices are either the quietest day available in the neighborhood of the event or an average of the monthly QQ days. The former choice was made here because data were available and because often none of the QQ days were very quiet. While averaging can eliminate much of the effects of disturbance, it is probably no better than using the quietest day available near the day for which Dst is being computed, since the average QQ day is substantially more disturbed than a very quiet day. The difference in the H component on a QQ day at 0000 and 2400 hours, which should be zero for a truly quiet day, can be as much as 20 γ, 10 γ being a not unusual value. As a result, differences between the empirical Dst and ground Dst of 10 γ are not necessarily significant.

This is illustrated clearly by the February 15 and 16, 1968, example. At 0000 UT between February 15 and 16, there is a jump in Dst of slightly more than 10 γ due simply to the difference in the quiet day at 2400 and 0000. This difference affects the fit over the next 6-8 hours as the empirical Dst, near zero at 0000 UT, rises slightly due to enhancement in dynamic pressure. If the empirical fit had been reinitialized to Dst = 0 at 0000 UT on February 16, the fit during the first 6 hours would have been much better. This problem can affect the fit between ground-determined Dst and the empirical Dst at all universal times.

The apparent dependence of the injection rate on the first power of the Y solar magnetospheric component of the interplanetary electric field is somewhat surprising. The Y component of the electric field is simply the radial solar wind velocity times the north-south component of the magnetic field and hence is the rate of transport of southward or 'mergible' flux to the magnetopause. The merging rate should be proportional to this rate, but it should also be proportional to the size of the merging region, which might also be expected to change with the IMF orientation. Perhaps the explanation of why the injection rate appears to be quadratically dependent on Ey for small Eyand linearly dependent on Eyfor large Ey is that the size of the merging region depends on Ey at first but then reaches a critical size. We note that in a preliminary stage of this analysis the source term was approximated by a term proportional to Ey2[Burton et al., 1973; Russell, 1974]. While small storms could be fit almost as well with the linear source term, we found that we predicted too large a storm when Eywas large.

Implicit in this discussion, of course, is the assumption that the ring current injection rate is linearly dependent on the merging rate. If it were not, the dependence of F(E) on Eywould in turn reflect this additional functional dependence. Another surprise is that the rectification of the interplanetary electric field appears to approximate that of a half-wave rectifier centered about a zero Y magnetospheric component. One expects merging to occur for even acute angles between the interplanetary and magnetospheric magnetic fields [cf. Sonnerup, 1974]. On the contrary, within the accuracy of this experiment this does not appear to be the case. Instead, the more common assumption about the nature of the solar wind interaction is supported [cf. Arnoldy, 1971 - Russell and McPherron, 1973 a, b].

In the classic storm sequence a sudden commencement is followed by the main phase decrease and finally by a recovery. This pattern was followed on many of the storms presented here but not on all, e.g., March 3 and May 1. Sudden commencements often occur without storms, and storms without sudden commencements [Akasofu, 1964].It is clear why. Sudden commencements are the results of enhancements in the solar wind dynamic pressure, while storm main phases are due to southward IMF's. Either phenomenon can occur without the other. However, during periods of enhanced pressure the IMF is generally larger than normal [Schatten, 1972], and as a result, so are any north-south components. When there is a southward component, ring current injection is then larger than normal. On the other hand, when the normally small IMF has a southward component, the resultant injection is not large enough to produce much decrease in Dst.

The fit to the sudden commencements that occurred in our sample of storms varied. In one case, i.e., February 23, 1967, the predicted sudden commencement was considerably larger than the observed sudden commencement recorded in the ground Dst. In other cases, the predicted sudden commencement is smaller, e.g., February 15, 1967 and 1968. For the rest the fit is quite good. Part of the difference may be due to the computation of the ground Dst. However, the major cause is probably the accuracy of the solar wind density and velocity measurements. We have compared hourly averages of dynamic pressure computed from hourly average density and velocity on Ogo 5 supplied by M. Neugebauer and Explorer 33 and 35 supplied by J. Binsack. The majority of the time, and especially during quiet times, the hourly average dynamic pressures agree within about 10-30%. However, occasionally, deviations up to 100% can occur, even during quiet times. The differences between simultaneous records of the two Explorer spacecraft were not unlike the differences between Explorer and Ogo. Hence these errors are not simply calibration errors. Rather, the ratio between the three instruments is essentially random, one is high at one time and low at another. As a result, deviations in empirical Dst and ground Dst during sudden commencements are often unavoidable. In addition, since these errors occur during quiet times as well as during disturbed times, the fit during all phases of a storm can be affected.

The end of the initial phase occurs when the Y component of the interplanetary electric field becomes sufficiently positive to result in ring current injection. In the storms studied, there was not necessarily a relaxation of pressure at this time. The same situation holds for storms without an initial phase. During the main phase the rate of Dst decrease is determined by the relative excess of ring current injection over decay. The main phase is seen to end in the storms studied when the injection rate drops, so that the decay rate exceeds the injection rate.

In a similar manner during the recovery phase the rate of change of Dst also depends on the relative amount of injection and decay. However, in two of our examples, February 17 and February 24, 1967, there is a slowing of the recovery with little positive Ey. The slowing on February 24 is slight and possibly not significant. However, on February 17 the recovery in observed Dst essentially stops around 0400 UT, while the predicted Dst continues to recover until the difference between the two is of the order of 20 γ. Possibly the discrepancy is caused by errors in the dynamic pressure, but possibly there is some additional physics that is not included in our formulation. One such possibility is the establishment of an energetic electron ring current inside the plasmapause. On the other hand, most of the variable recovery rates are explained by variations in the injection rate. On occasion this strengthening of the injection rate during a recovery phase can lead to a new main phase. Examples of this are February 7, 8, 1967, and February 27-29, 1968. We see no evidence in these data that two recovery rates are present as a general rule during storms.

Russell et al. [ 1974] have reported an apparent threshold for storms possibly dependent on the level of Dst. Within our model, such a threshold arises because injection does not occur unless F(E) exceeds aDst0, and hence our model is in agreement with their observations. We note that while in theory the condition F(E)>aDst0 will generate an injection, in practice, F(E)>>aDst0 during an injection. Further, the amount of injection appears to be limited as much by the duration of the enhanced injection as by its magnitude. For example, while Ey= 3 mV m-1 can maintain Dst at - 100 γ, it requires 14 hours to reach -90 γ starting from 0 γ. However, - 100 γ can be reached in only 5 hours with a 5 mV m-1 electric field. A corollary of this is that an injection rate during the recovery phase that is only a small fraction of that during the main phase can significantly slow the recovery rate. For example, a dawn-dusk field of 1.5-2 mV m-1 will maintain Dst near -40 γ, while 1 mV m-1 will cut the recovery rate by half.

The storms studied here included main phase minima down to about - 120 γ. However, storms during the period for which Dst values are published include some with minima as low as -400 γ. Although no simultaneous interplanetary measurements are available for these larger events, it is interesting to speculate on the magnitude of the dawn-dusk electric field necessary to generate them. Since the solar wind conditions are not known, simplifying assumptions must be made. If the dynamic pressure is taken to be constant, the hourly ring current injection rate is F(E) ~ ΔDst + ring current decay. For example, if Dst dropped from - 100 to - 150 γ in 1 hour, ΔDst = -50 γ, and the ring current decay is Dst/8 ~ 12 γ. Thus the required injection rate is 62 hour-1, which would require an average dawn-dusk field for the hour of about 12 mV/m. For V = 500 km/s this would mean that Bz = -24 γ.

For the storms with minima below -300 γ, peak hourly injection rates of 100-130 γ/hour occurred, injection rates exceeding 50 γ/hour being sustained for periods of 5-8 hours. These injection rates require peak hourly average dawn-dusk electric fields of 20-25 mV/m and sustained fields in excess of 10 mV/m. If it is assumed that the solar wind velocities ranged from 500 to 1000 km/s, peak hourly average Bzof 20-50 γ and sustained fields of 10-20 γ would be required.

In our sample, covering 1967 and 1968, a dawn-dusk electric field of about 9 mV/m with a solar wind velocity of 650 km/s was sustained for about I hour on February 7, 1967. On February 16, 1967, the dawn-dusk electric field exceeded 15 mV/m for about 30 min, and it peaks at about 18 mV/m, with a solar wind velocity of about 600 km/s. The super storms, mostly occurring in 1957 and 1958, quite possibly occurred when the solar wind velocity was higher, of the order of 1000 km/s. It is also not unreasonable that larger southward magnetic fields than those seen here occur. However, even the biggest storms do not require larger southward IMF's than those seen on occasion during 1967 and 1968. These large fields would have to be maintained over 5-8 hours as opposed to the 30 min to I hour seen in the storms presented here. If one accepts this possibility, it appears that even the largest storms are not inconsistent with our model.

Acknowledgments. We are indebted to J. Binsack for the Explorer 33 and 35 MIT plasma data and to C. P. Sonett and D. S. Colburn for the Explorer 33 and 35 magnetic field data. The World Data Center A for Geomagnetism provided the digitized ground magnetograms. We are indebted to C. R. Clauer for producing from these records the 2.5-min Dst values, which were essential to this study. This work was supported in part by NSF grant GA 34148X and ONR grant N00014-69A-0200-4016. Institute of Geophysics and Planetary Physics publication 1329-51.

The Editor thanks R. L. Arnoldy and G. Rostoker for their assistance in evaluating this paper.